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14 votes
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Calculating alpha and its meaning

Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas)...
Matthew Gunn's user avatar
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12 votes
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Fama-French factor model: why mimicking portfolios?

The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-...
Matthew Gunn's user avatar
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8 votes
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Why do anomalies disappear after they get detected?

The best explanation I have seen so far is the so-called Adaptive Market Hypothesis by Andrew Lo: The adaptive market hypothesis, as proposed by Andrew Lo, is an attempt to reconcile economic ...
vonjd's user avatar
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8 votes
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How are modern portfolio theory (MPT) and CAPM related?

CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only ...
MGL's user avatar
  • 516
8 votes

Closed-form analytical solution for the variance of the minimum-variance portfolio?

A few more steps beyond your last equation gives the answer. With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have $$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{...
RRL's user avatar
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7 votes
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Marginal Risk Contribution Formula

concerning your first question: the derivative does not disappear: $\sigma(R_p)$ contains the square root. To be more precise, set $$ \sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)...
Cettt's user avatar
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7 votes

What’s the derivative of the sharpe ratio for one asset? Trying to optimize on it for a model

I agree that the paper could be much clearer: what it calls the “Sharp ratio derivative” is actually the “differential Sharpe ratio” proposed in a NIPS paper by Moody & Safell. In Section 2.2 of ...
A. G.'s user avatar
  • 200
7 votes

Most significant research articles for practical investors with research perspectives

A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work. Here are a few ...
Alex C's user avatar
  • 9,382
7 votes

Portfolio Optimization and Global Minimum Variance Portfolio (GMV)

1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. ...
demully's user avatar
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7 votes

Contribution of an asset's variance to portfolio variance

In this answer, I am assuming that you want to keep correlations constant. To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written ...
Kermittfrog's user avatar
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7 votes
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Why isn't the asset with minimum variance given a 100% portfolio weight?

Diversification is key. The clear cut answer is diversification. A weighted combination of assets will more often than not show a lower return variance than even the asset with the lowest variance ...
Kermittfrog's user avatar
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6 votes
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Maximum Certainty Equivalent Portfolio with Transaction Costs

Seems like a small mistake in the last equation. It should read $\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\...
krise's user avatar
  • 116
6 votes
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Generalized Mean Variance Portfolio

Something to perhaps realize is that your two problems may not be as different as you think if $\lambda$ is an ad-hoc parameter. For any solution to your 2nd problem (where $\theta > 1$), there ...
Matthew Gunn's user avatar
  • 6,954
6 votes
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Rockafellar-Uryasev mean-CVaR optimiztion

$VaR_\alpha$ is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called $\alpha\in R$. (C.f., the program described in Theorem 2 of that paper, or ...
Drew's user avatar
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6 votes
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Portfolio Optimization sum of weights constraint with short selling

In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), ...
nbbo2's user avatar
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6 votes
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Does mean variance optimization work in real life? If so, why are defined benefit pension funds so underfunded?

You are asking two questions: whether MVO works in real life and portfolio managers actually use it? whether defined benefit schemes use this tool? Concerning 1. the answer is generally no, although ...
Sebapi's user avatar
  • 460
5 votes

Which algorithms do robo-advisors use?

Well, I did some modest research on this topic, looking at peers. Most of them use Modern Portfolio Theory, see this pic: You can find this small survey here: https://www.linkedin.com/pulse/...
RockZen's user avatar
  • 51
5 votes
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how can we know the residual return will be uncorrelated with the market return

Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS ...
Richi Wa's user avatar
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5 votes
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Why did Markowitz not derive an equation for the efficient frontier?

It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the ...
Alex C's user avatar
  • 9,382
5 votes

Backtest Results needed to Model Validate my Modern Portfolio Theory model

There is a recent a paper recently using a population test of all CRSP data from 1925-2013 as a test of whether a mean and a variance exist versus they do not exist. It overwhelmingly excluded mean-...
Dave Harris's user avatar
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5 votes
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Optimisation with strong correlated Assets

This answer will try and outline all the different possibilities I came across over the last couple of years, including drawbacks. But first, let me outline the problem a little. To appreciate the ...
vanguard2k's user avatar
  • 2,915
5 votes

Why do anomalies disappear after they get detected?

Any anomaly that can be phrased as a "mispricing" or "relative value" opportunity can be expected to disappear as more people discover it and trade on it. For example, say that stock movements over ...
Chris Taylor's user avatar
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5 votes
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Finding a minimum variance portfolio when using a regulariser?

You're not going to get an analytic formula except in special cases of function $\rho(x)$. And you're probably going to want $\rho$ convex. If $\rho$ is convex, the problem is a convex optimization ...
Matthew Gunn's user avatar
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5 votes
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Widely accepted methods for coming up with the co-variance matrix of assets?

Multivariate volatility models for replacing the sample covariance matrix with in the mean-variance portfolio selection model: RiskMetrics 1996 EWMA (Exponentially weighted moving average) covariance ...
develarist's user avatar
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5 votes

SDF as an affine transformation of the tangency portfolio

Coming back to my question after I replicated the paper for my thesis, where I found that my resulting SDF is always strictly positive and hovering around the value 1, just as expected given the ...
ln_greenspan's user avatar
5 votes
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Asset Allocation with near zero rates

Many pension funds use projected asset class returns (capital market assumptions or CMAs) and backward-looking estimates of volatilities and correlations to set the strategic asset allocation. A 10-...
RRL's user avatar
  • 3,680
5 votes

Asset Allocation with near zero rates

I'll add some comments, recognizing that 1) they are highly opinionated, and 2) they don't actually offer any real solutions. Hopefully more thoughtful and useful answers will emerge. First of all, ...
Helin's user avatar
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5 votes
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Closed-form analytical solution for the variance of the minimum-variance portfolio?

Let \begin{align} a&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}\\ b&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\boldsymbol{\mu}\\ c&\equiv \boldsymbol{\mu}^T\mathbf{\Sigma}^{-1}\...
Kermittfrog's user avatar
  • 6,663
5 votes
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Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations

Let's stick with the nomenclature in the literature and let $\gamma$ denote the decision maker's risk aversion coefficient. The optimization problem is $$ \max_{\mathrm{w}} \mathrm{w}^T\mathrm{\mu}-\...
Kermittfrog's user avatar
  • 6,663
5 votes
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Maximum skewness portfolio solution derived from its Lagrangean formulation

Unfortunately, there exist no closed form for this. The Lagrangean reads $$ L(w,\lambda)=w^TM_3(w\otimes w)-\lambda(w^T\mathbf{1}-1) $$ with first order conditions $$ \begin{align} \frac{\partial L }{\...
Kermittfrog's user avatar
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